CauchyintegraltheerenTheorem(3.2.2)Letf(z)beanalyticonasimpleconnecteddomainD,and letCbeasimpleclosedcurveonD.Then6. f(z)dz = 0shaUni.ofSci&TechFCV&ITNovember5,201918/53angWa
Cauchy integral theorem Theorem (3.2.2) Let f(z) be analytic on a simple connected domain D, and let C be a simple closed curve on D. Then I C f(z)dz = 0 C D Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 18 / 53
auchyntsralthaerenTheorem(3.2.3)Supposethatfisanalyticonasimpleconnected domainD.Then,foranytwocurvesCiandC2joiningtwopointszoandz1,wehave[f(z)dz = /(2z)dzSolution.Letus considerC=C2+C.ThenCisaclosedcurve.ByTheorem3.2.1,thus0 = / f(z)dzC1= /, f(2)dz + /~f(2)dzC2= Je f(2)dz - J, (2)dzTherefore,wegettheconclusion.November 5, 201919/53of Sci &Tech)FCV&IT
Cauchy integral theorem Theorem (3.2.3) Suppose that f is analytic on a simple connected domain D. Then, for any two curves C1 and C2 joining two points z0 and z1, we have Z C1 f(z)dz = Z C2 f(z)dz z0 z1 C2 C1 Solution. Let us consider C = C2 + C − 1 . Then C isa closed curve. By Theorem 3.2.1, thus 0 = Z C f(z)dz = Z C2 f(z)dz + Z C − 1 f(z)dz = Z C2 f(z)dz − Z C1 f(z)dz Therefore, we get the conclusion. Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 19 / 53